10.1.ipynb

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    "# Chapter 10: Parametric Equations and Polar Coordinates\n",
    "\n",
    "## Section 10.1: Curves Defined by Parametric Equations\n",
    "\n",
    "In this section, we learn about parametrically defined curved, i.e. curves of the form:\n",
    "$$x = f(t)\\quad y = g(t)$$\n",
    "Our first goal is to simply familiarize ourselves with what such curves look like.\n",
    "First of all, we can use parametric curves to draw the curves that come from functions.\n",
    "For such curves, we will choose $x = t$ so that $y = g(t) = g(x)$ is just the graph of a function.\n",
    "In the code block below, feel free to edit the function definition.\n",
    "The code is written in Julia."
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    "using Plots"
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    "# The values at which to evaluate the parametric function. The syntax is  smallest:step:largest.\n",
    "t = -10:.1:10;\n",
    "x = t;\n",
    "\n",
    "# This is the function definition. Feel free to change this to anything you feel like! \n",
    "# You might need to first learn how to express your function correctly in Julia syntax.\n",
    "y = t.^3+1;\n",
    "\n",
    "# Finally, plot the functions\n",
    "\n",
    "plot(x, y)\n",
    "plot!(title= \"A Simple Parametric Curve\", ylabel= \"y\", xlabel=\"x\", legend=false)\n",
    "plot!(ylim=[-1, 1]*maximum(abs.(y)))"
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    "We can make things slightly more interesting by instead setting $y=t$ and then $x=f(t)=f(y)$, so that we have just turned our typical functions on their side."
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    "# The values at which to evaluate the parametric function. The syntax is  smallest:step:largest.\n",
    "t = -10:.1:10;\n",
    "y = t;\n",
    "\n",
    "# This is the function definition. Feel free to change this to anything you feel like! \n",
    "# You might need to first learn how to express your function correctly in Julia syntax.\n",
    "x = sin.(y);\n",
    "\n",
    "# Finally, plot the functions\n",
    "\n",
    "plot(x, y)\n",
    "plot!(title= \"A Simple Parametric Curve\", ylabel= \"y\", xlabel=\"x\", legend=false)\n",
    "plot!(xlim=[-1, 1]*maximum(abs.(x)))"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But parametric curves are much more interesting than curves that have to pass either the vertical line test or the horizontal line test!\n",
    "We can very easily create parametric curves that trace out a circle, ellipse, and much more.\n",
    "Here's some examples of the interesting curves we can get with this quite powerful tool:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 1\n",
    "t = 0:0.01:2*pi;\n",
    "r = 1;\n",
    "x = r*cos.(t);\n",
    "y = r*sin.(t);\n",
    "plot(x,y)\n",
    "plot!(title= \"A Non-function Parametric Curve\", xlabel = \"x\", ylabel = \"y\", legend = false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "<g clip-path=\"url(#clip7000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:end;\" transform=\"rotate(0, 55.9406, 77.0717)\" x=\"55.9406\" y=\"77.0717\">0.75</text>\n",
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       "<g clip-path=\"url(#clip7000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:21; text-anchor:middle;\" transform=\"rotate(0, 321.128, 18)\" x=\"321.128\" y=\"18\">A Non-function Parametric Curve</text>\n",
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       "  579.02,163.163 578.748,159.949 578.451,156.75 578.128,153.566 577.779,150.398 577.405,147.249 577.004,144.12 576.579,141.011 576.127,137.924 575.651,134.86 \n",
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       "  568.739,103.029 567.96,100.335 567.157,97.6798 566.33,95.0636 565.478,92.4878 564.601,89.9533 563.7,87.4612 562.775,85.0123 561.826,82.6078 560.852,80.2486 \n",
       "  559.855,77.9356 558.834,75.6697 557.789,73.4519 556.72,71.2831 555.628,69.164 554.512,67.0956 553.373,65.0787 552.211,63.1141 551.025,61.2025 549.817,59.3448 \n",
       "  548.586,57.5417 547.332,55.7939 546.055,54.1021 544.756,52.4669 543.435,50.8891 542.091,49.3693 540.725,47.908 539.338,46.5059 537.928,45.1635 536.497,43.8814 \n",
       "  535.044,42.6601 533.57,41.5 532.075,40.4016 530.558,39.3653 529.021,38.3917 527.462,37.481 525.884,36.6337 524.284,35.85 522.665,35.1303 521.025,34.4748 \n",
       "  519.365,33.8839 517.685,33.3578 515.986,32.8966 514.267,32.5006 512.529,32.1699 510.772,31.9047 508.996,31.705 507.201,31.571 505.387,31.5027 503.555,31.5 \n",
       "  501.705,31.5632 499.837,31.692 497.951,31.8864 496.047,32.1464 494.126,32.4719 492.187,32.8627 490.231,33.3187 488.259,33.8396 486.269,34.4254 484.263,35.0757 \n",
       "  482.241,35.7903 480.203,36.5689 478.149,37.4112 476.079,38.3169 473.993,39.2855 471.893,40.3168 469.777,41.4102 467.646,42.5654 465.501,43.7819 463.341,45.0592 \n",
       "  461.167,46.3968 458.979,47.7942 456.777,49.2508 454.562,50.766 452.334,52.3392 450.092,53.9698 447.837,55.6571 445.57,57.4005 443.29,59.1992 440.998,61.0526 \n",
       "  438.694,62.9599 436.378,64.9203 434.051,66.9331 431.712,68.9975 429.363,71.1125 427.002,73.2774 424.631,75.4914 422.25,77.7534 419.859,80.0627 417.457,82.4183 \n",
       "  415.046,84.8192 412.626,87.2645 410.196,89.7533 407.758,92.2845 405.311,94.857 402.855,97.47 400.391,100.122 397.92,102.813 395.441,105.541 392.954,108.304 \n",
       "  390.46,111.103 387.959,113.936 385.452,116.802 382.938,119.699 380.417,122.626 377.891,125.583 375.36,128.568 372.822,131.58 370.28,134.617 367.733,137.679 \n",
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       "  130.005,359.492 128.263,359.172 126.542,358.786 124.839,358.335 123.156,357.82 121.493,357.239 119.85,356.594 118.228,355.884 116.625,355.111 115.043,354.274 \n",
       "  113.481,353.373 111.941,352.409 110.421,351.383 108.922,350.294 107.444,349.144 105.988,347.933 104.554,346.66 103.141,345.327 101.749,343.935 100.38,342.483 \n",
       "  99.0329,340.972 97.7079,339.404 96.4053,337.777 95.1251,336.095 93.8675,334.356 92.6327,332.561 91.4207,330.712 90.2317,328.809 89.0657,326.853 87.923,324.844 \n",
       "  86.8036,322.784 85.7076,320.673 84.6352,318.512 83.5864,316.302 82.5613,314.044 81.5602,311.738 80.5829,309.386 79.6298,306.989 78.7008,304.547 77.796,302.062 \n",
       "  76.9155,299.534 76.0595,296.965 75.228,294.355 74.4211,291.706 73.6388,289.018 72.8813,286.293 72.1486,283.532 71.4408,280.736 70.758,277.906 70.1003,275.043 \n",
       "  69.4676,272.149 68.8601,269.223 68.2778,266.269 67.7208,263.286 67.1891,260.277 66.6829,257.241 66.202,254.181 65.7467,251.098 65.3169,247.992 64.9127,244.866 \n",
       "  64.5341,241.72 64.1812,238.555 63.854,235.374 63.5524,232.177 63.2767,228.965 63.0267,225.74 62.8026,222.503 62.6043,219.255 62.4318,215.998 62.2852,212.732 \n",
       "  62.1645,209.46 62.0696,206.183 62.0007,202.901 61.9577,199.617 61.9406,196.331 61.9495,193.045 61.9842,189.76 62.0449,186.477 62.1314,183.198 62.2439,179.924 \n",
       "  62.3823,176.657 62.5465,173.397 62.7366,170.146 62.9525,166.906 63.1943,163.677 63.4618,160.46 63.7551,157.258 64.0742,154.072 64.4189,150.902 64.7893,147.75 \n",
       "  65.1854,144.617 65.607,141.505 66.0542,138.414 66.5269,135.346 67.0251,132.303 67.5487,129.285 68.0976,126.294 68.6719,123.33 69.2713,120.395 69.896,117.491 \n",
       "  70.5458,114.618 71.2206,111.777 71.9205,108.97 72.6452,106.198 73.3948,103.461 74.1692,100.762 74.9683,98.1001 75.792,95.4776 76.6402,92.8953 77.5129,90.3542 \n",
       "  78.4099,87.8552 79.3312,85.3994 80.2767,82.9878 81.2463,80.6213 82.2398,78.3009 83.2573,76.0274 84.2985,73.8019 85.3634,71.6252 86.4519,69.4982 87.5639,67.4216 \n",
       "  88.6992,65.3965 89.8578,63.4235 91.0395,61.5034 92.2442,59.6371 93.4717,57.8252 94.7221,56.0685 95.9951,54.3677 97.2906,52.7235 98.6085,51.1366 99.9486,49.6075 \n",
       "  101.311,48.1368 102.695,46.7253 104.101,45.3733 105.529,44.0816 106.978,42.8505 108.449,41.6806 109.941,40.5724 111.454,39.5262 112.988,38.5426 114.543,37.6218 \n",
       "  116.119,36.7644 117.715,35.9705 119.331,35.2406 120.968,34.5749 122.625,33.9737 124.301,33.4372 125.997,32.9657 127.713,32.5593 129.448,32.2182 131.202,31.9426 \n",
       "  132.975,31.7324 134.767,31.5879 136.578,31.5091 138.407,31.4961 140.254,31.5487 142.12,31.667 144.003,31.851 145.904,32.1006 147.823,32.4157 149.758,32.7961 \n",
       "  151.711,33.2417 153.681,33.7523 155.668,34.3278 157.671,34.9678 159.691,35.6722 161.727,36.4406 163.779,37.2728 165.846,38.1684 167.929,39.127 170.027,40.1483 \n",
       "  172.141,41.2319 174.269,42.3773 176.412,43.5841 178.569,44.8518 180.741,46.1798 182.927,47.5677 185.126,49.0149 187.339,50.5208 189.566,52.0848 191.806,53.7063 \n",
       "  194.058,55.3846 196.324,57.1191 198.601,58.9091 200.891,60.7538 203.194,62.6526 205.507,64.6046 207.833,66.6091 210.17,68.6653 212.517,70.7723 214.876,72.9293 \n",
       "  217.246,75.1355 219.625,77.39 222.015,79.6918 224.415,82.0401 226.824,84.4338 229.243,86.8722 231.671,89.3541 234.109,91.8785 236.554,94.4446 239.008,97.0512 \n",
       "  241.471,99.6972 243.941,102.382 246.419,105.104 248.905,107.862 251.398,110.655 253.897,113.483 256.404,116.343 258.917,119.235 261.436,122.158 263.961,125.11 \n",
       "  266.492,128.091 269.028,131.098 271.57,134.132 274.116,137.19 276.668,140.271 279.223,143.375 281.783,146.5 284.347,149.644 286.914,152.807 289.485,155.988 \n",
       "  292.059,159.184 294.636,162.394 297.216,165.618 299.798,168.855 302.382,172.102 304.968,175.358 307.555,178.623 310.144,181.894 312.734,185.171 315.325,188.452 \n",
       "  317.917,191.737 320.508,195.023 323.1,198.309 325.692,201.594 328.283,204.877 330.874,208.156 333.463,211.43 336.051,214.698 338.638,217.959 341.223,221.211 \n",
       "  343.806,224.452 346.387,227.682 348.965,230.899 351.541,234.103 354.113,237.291 356.682,240.462 359.248,243.615 361.809,246.75 364.367,249.864 366.921,252.956 \n",
       "  369.469,256.026 372.013,259.071 374.552,262.091 377.086,265.085 379.613,268.051 382.136,270.987 384.651,273.894 387.161,276.77 389.664,279.613 392.16,282.423 \n",
       "  394.649,285.198 397.131,287.938 399.605,290.64 402.071,293.305 404.529,295.93 406.979,298.516 409.42,301.06 411.853,303.563 414.276,306.022 416.69,308.437 \n",
       "  419.095,310.807 421.489,313.131 423.874,315.409 426.248,317.638 428.612,319.819 430.965,321.95 433.307,324.03 435.638,326.06 437.958,328.037 440.265,329.961 \n",
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       "  561.518,309.764 562.475,307.374 563.408,304.939 564.317,302.461 565.201,299.939 566.061,297.377 566.896,294.773 567.707,292.13 568.494,289.449 569.255,286.73 \n",
       "  569.992,283.974 570.703,281.184 571.39,278.359 572.052,275.501 572.689,272.612 573.3,269.691 573.886,266.741 574.447,263.763 574.983,260.758 575.493,257.726 \n",
       "  575.978,254.67 576.438,251.59 576.872,248.488 577.28,245.365 577.663,242.222 578.02,239.06 578.351,235.882 578.657,232.687 578.936,229.477 579.19,226.254 \n",
       "  579.419,223.019 579.621,219.773 579.798,216.517 579.948,213.253 580.073,209.982 580.172,206.705 580.245,203.424 580.292,200.14 580.314,196.854 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 2\n",
    "t = 0:0.01:2*pi;\n",
    "r = 1;\n",
    "x = r*cos.(t);\n",
    "y = r*sin.(2*t);\n",
    "plot(x,y)\n",
    "plot!(title= \"A Non-function Parametric Curve\", xlabel = \"x\", ylabel = \"y\", legend = false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "  235.578,239.165 237.404,237.874 239.258,236.551 241.138,235.202 243.039,233.83 244.957,232.439 246.889,231.033 248.83,229.618 250.777,228.196 252.725,226.773 \n",
       "  254.67,225.354 256.608,223.941 258.536,222.54 260.448,221.156 262.342,219.791 264.213,218.451 266.057,217.14 267.871,215.861 269.651,214.619 271.393,213.416 \n",
       "  273.094,212.258 274.75,211.146 276.358,210.085 277.915,209.078 279.417,208.127 280.862,207.235 282.247,206.405 283.569,205.639 284.826,204.938 286.015,204.305 \n",
       "  287.135,203.742 288.182,203.249 289.157,202.827 290.056,202.478 290.878,202.202 291.623,201.998 292.289,201.868 292.875,201.811 293.381,201.826 293.805,201.913 \n",
       "  294.149,202.07 294.412,202.297 294.593,202.591 294.694,202.951 294.715,203.375 294.657,203.86 294.52,204.405 294.306,205.006 294.016,205.66 293.652,206.365 \n",
       "  293.215,207.117 292.707,207.913 292.131,208.749 291.489,209.621 290.783,210.526 290.015,211.46 289.188,212.418 288.306,213.396 287.371,214.39 286.387,215.396 \n",
       "  285.356,216.409 284.282,217.425 283.169,218.44 282.02,219.449 280.838,220.448 279.629,221.432 278.395,222.398 277.14,223.341 275.868,224.256 274.584,225.141 \n",
       "  273.291,225.99 271.993,226.801 270.695,227.569 269.4,228.291 268.113,228.964 266.837,229.584 265.576,230.149 264.336,230.656 263.118,231.102 261.928,231.485 \n",
       "  260.769,231.803 259.644,232.054 258.558,232.236 257.514,232.348 256.516,232.389 255.565,232.357 254.667,232.253 253.823,232.076 253.038,231.826 252.313,231.503 \n",
       "  251.651,231.107 251.054,230.639 250.526,230.101 250.068,229.492 249.682,228.816 249.369,228.073 249.132,227.265 248.972,226.395 248.89,225.465 248.887,224.477 \n",
       "  248.964,223.435 249.121,222.341 249.359,221.199 249.678,220.012 250.079,218.784 250.56,217.517 251.122,216.217 251.764,214.887 252.485,213.531 253.284,212.153 \n",
       "  254.16,210.757 255.112,209.348 256.138,207.931 257.236,206.508 258.405,205.086 259.642,203.668 260.944,202.258 262.311,200.862 263.738,199.482 265.223,198.124 \n",
       "  266.764,196.792 268.357,195.489 269.999,194.22 271.686,192.988 273.416,191.798 275.185,190.652 276.989,189.555 278.825,188.509 280.688,187.517 282.576,186.582 \n",
       "  284.483,185.707 286.406,184.895 288.342,184.147 290.285,183.465 292.233,182.851 294.18,182.307 296.123,181.834 298.058,181.433 299.981,181.104 301.888,180.848 \n",
       "  303.775,180.666 305.637,180.556 307.472,180.519 309.274,180.554 311.042,180.661 312.771,180.838 314.457,181.083 316.097,181.396 317.688,181.774 319.227,182.216 \n",
       "  320.71,182.718 322.135,183.278 323.499,183.895 324.8,184.564 326.034,185.282 327.2,186.047 328.296,186.854 329.319,187.701 330.268,188.583 331.142,189.496 \n",
       "  331.938,190.437 332.656,191.401 333.296,192.384 333.855,193.382 334.333,194.39 334.731,195.405 335.047,196.421 335.282,197.435 335.437,198.441 335.511,199.437 \n",
       "  335.505,200.416 335.42,201.376 335.257,202.311 335.017,203.218 334.702,204.093 334.313,204.932 333.853,205.731 333.322,206.487 332.723,207.196 332.059,207.854 \n",
       "  331.332,208.459 330.544,209.008 329.698,209.498 328.798,209.927 327.846,210.292 326.846,210.591 325.8,210.823 324.712,210.985 323.587,211.077 322.426,211.098 \n",
       "  321.235,211.046 320.017,210.922 318.775,210.724 317.514,210.454 316.238,210.11 314.95,209.694 313.655,209.206 312.357,208.648 311.059,208.021 309.767,207.325 \n",
       "  308.483,206.564 307.212,205.738 305.957,204.851 304.724,203.905 303.515,202.902 302.335,201.845 301.187,200.737 300.075,199.582 299.003,198.383 297.974,197.143 \n",
       "  296.991,195.867 296.058,194.558 295.178,193.22 294.353,191.857 293.588,190.474 292.883,189.074 292.243,187.663 291.67,186.243 291.165,184.821 290.73,183.399 \n",
       "  290.369,181.983 290.082,180.576 289.87,179.184 289.736,177.81 289.681,176.459 289.704,175.135 289.808,173.841 289.993,172.582 290.258,171.362 290.605,170.183 \n",
       "  291.032,169.051 291.541,167.967 292.13,166.936 292.798,165.96 293.546,165.042 294.371,164.184 295.273,163.389 296.25,162.66 297.3,161.997 298.422,161.403 \n",
       "  299.614,160.879 300.873,160.425 302.198,160.044 303.585,159.736 305.032,159.501 306.536,159.339 308.095,159.249 309.704,159.233 311.362,159.288 313.064,159.415 \n",
       "  314.808,159.611 316.589,159.875 318.404,160.207 320.249,160.603 322.121,161.061 324.016,161.58 325.929,162.157 327.857,162.788 329.795,163.471 331.741,164.203 \n",
       "  333.688,164.98 335.635,165.799 337.576,166.656 339.507,167.547 341.425,168.468 343.325,169.416 345.204,170.386 347.058,171.373 348.882,172.375 350.674,173.385 \n",
       "  352.428,174.401 354.143,175.417 355.814,176.429 357.438,177.432 359.012,178.424 360.532,179.398 361.997,180.351 363.401,181.279 364.744,182.178 366.023,183.043 \n",
       "  367.235,183.871 368.377,184.659 369.449,185.401 370.447,186.096 371.371,186.74 372.219,187.33 372.989,187.863 373.681,188.336 374.293,188.747 374.825,189.093 \n",
       "  375.276,189.374 375.647,189.586 375.936,189.729 376.144,189.801 376.271,189.802 376.318,189.73 376.286,189.585 376.174,189.367 375.986,189.076 375.72,188.712 \n",
       "  375.38,188.276 374.967,187.768 374.483,187.19 373.929,186.543 373.308,185.829 372.622,185.05 371.874,184.207 371.066,183.303 370.202,182.34 369.284,181.322 \n",
       "  368.315,180.25 367.299,179.129 366.239,177.961 365.139,176.75 364.001,175.5 362.83,174.214 361.629,172.897 360.402,171.552 359.154,170.183 357.887,168.794 \n",
       "  356.607,167.391 355.316,165.977 354.019,164.556 352.721,163.133 351.425,161.712 350.134,160.298 348.854,158.895 347.588,157.508 346.341,156.14 345.115,154.796 \n",
       "  343.916,153.48 342.746,152.195 341.61,150.947 340.511,149.738 339.453,148.572 338.439,147.453 337.472,146.384 336.556,145.368 335.695,144.408 334.89,143.506 \n",
       "  334.144,142.666 333.461,141.889 332.843,141.178 332.293,140.534 331.811,139.96 331.401,139.455 331.064,139.022 330.803,138.662 330.617,138.374 330.509,138.159 \n",
       "  330.48,138.017 330.531,137.948 330.662,137.952 330.873,138.027 331.166,138.174 331.54,138.389 331.995,138.672 332.53,139.022 333.146,139.436 333.841,139.912 \n",
       "  334.615,140.447 335.466,141.039 336.394,141.685 337.395,142.382 338.47,143.127 339.616,143.916 340.831,144.746 342.112,145.613 343.458,146.513 344.865,147.442 \n",
       "  346.332,148.397 347.855,149.372 349.431,150.364 351.057,151.368 352.73,152.38 354.447,153.396 356.203,154.412 357.996,155.422 359.822,156.423 361.677,157.41 \n",
       "  363.557,158.379 365.458,159.325 367.376,160.245 369.308,161.135 371.25,161.99 373.196,162.807 375.144,163.582 377.089,164.312 379.027,164.993 380.955,165.622 \n",
       "  382.867,166.196 384.761,166.712 386.631,167.168 388.475,167.561 390.289,167.89 392.069,168.151 393.81,168.344 395.511,168.468 397.166,168.52 398.774,168.5 \n",
       "  400.33,168.408 401.832,168.242 403.276,168.004 404.661,167.692 405.982,167.308 407.239,166.851 408.427,166.324 409.546,165.727 410.593,165.061 411.567,164.328 \n",
       "  412.465,163.531 413.287,162.671 414.031,161.75 414.696,160.771 415.282,159.738 415.787,158.652 416.211,157.517 416.554,156.337 416.815,155.115 416.996,153.854 \n",
       "  417.097,152.559 417.117,151.233 417.058,149.881 416.92,148.506 416.705,147.113 416.415,145.706 416.05,144.29 415.613,142.868 415.104,141.445 414.528,140.026 \n",
       "  413.885,138.615 413.178,137.216 412.409,135.833 411.583,134.472 410.7,133.135 409.764,131.827 408.78,130.553 407.748,129.315 406.674,128.118 405.561,126.965 \n",
       "  404.411,125.859 403.23,124.805 402.02,123.804 400.785,122.86 399.53,121.975 398.259,121.153 396.974,120.394 395.681,119.702 394.383,119.077 393.085,118.522 \n",
       "  391.79,118.038 390.503,117.625 389.227,117.285 387.967,117.017 386.727,116.823 385.509,116.702 384.319,116.653 383.161,116.677 382.036,116.772 380.951,116.938 \n",
       "  379.907,117.173 378.909,117.475 377.959,117.843 377.061,118.274 376.218,118.767 375.433,119.318 374.708,119.926 374.047,120.587 373.451,121.297 372.923,122.055 \n",
       "  372.466,122.856 372.08,123.697 371.769,124.573 371.532,125.482 371.373,126.418 371.291,127.379 371.289,128.359 371.366,129.355 371.524,130.362 371.763,131.376 \n",
       "  372.083,132.392 372.485,133.407 372.967,134.415 373.529,135.412 374.172,136.394 374.894,137.357 375.694,138.297 376.57,139.209 377.523,140.089 378.549,140.934 \n",
       "  379.648,141.74 380.817,142.503 382.055,143.219 383.358,143.886 384.725,144.499 386.153,145.058 387.639,145.557 389.18,145.996 390.773,146.371 392.415,146.681 \n",
       "  394.103,146.923 395.834,147.097 397.603,147.2 399.407,147.233 401.243,147.192 403.107,147.08 404.994,146.894 406.902,146.635 408.825,146.303 410.761,145.898 \n",
       "  412.704,145.422 414.652,144.875 416.599,144.258 418.542,143.573 420.477,142.822 422.4,142.007 424.307,141.129 426.193,140.192 428.055,139.198 429.89,138.149 \n",
       "  431.692,137.05 433.459,135.902 435.188,134.71 436.873,133.476 438.513,132.206 440.104,130.901 441.642,129.568 443.125,128.209 444.549,126.829 445.913,125.431 \n",
       "  447.213,124.021 448.447,122.603 449.612,121.18 450.707,119.758 451.73,118.341 452.678,116.932 453.551,115.538 454.347,114.16 455.064,112.805 455.703,111.476 \n",
       "  456.261,110.178 456.739,108.913 457.136,107.686 457.451,106.501 457.686,105.361 457.84,104.269 457.913,103.23 457.906,102.244 457.821,101.317 457.657,100.45 \n",
       "  457.417,99.6446 457.101,98.9045 456.712,98.2309 456.25,97.6257 455.719,97.0902 455.119,96.6258 454.455,96.2332 453.727,95.9132 452.938,95.6662 452.092,95.4924 \n",
       "  451.192,95.3916 450.239,95.3635 449.238,95.4075 448.192,95.5226 447.104,95.7078 445.978,95.9616 444.818,96.2824 443.626,96.6682 442.408,97.117 441.166,97.6265 \n",
       "  439.905,98.194 438.629,98.8168 437.341,99.4919 436.046,100.216 434.747,100.986 433.45,101.799 432.157,102.65 430.873,103.536 429.602,104.452 428.348,105.396 \n",
       "  427.115,106.363 425.907,107.348 424.727,108.347 423.579,109.357 422.468,110.372 421.396,111.388 420.367,112.401 419.384,113.406 418.452,114.4 417.572,115.377 \n",
       "  416.748,116.334 415.983,117.266 415.279,118.17 414.64,119.041 414.067,119.875 413.562,120.669 413.129,121.419 412.768,122.122 412.481,122.774 412.271,123.373 \n",
       "  412.137,123.915 412.082,124.397 412.107,124.818 412.211,125.175 412.396,125.467 412.663,125.69 413.01,125.844 413.438,125.928 413.947,125.94 414.537,125.879 \n",
       "  415.206,125.746 415.955,125.539 416.781,125.26 417.683,124.907 418.661,124.483 419.712,123.987 420.834,123.42 422.027,122.784 423.286,122.081 424.611,121.311 \n",
       "  425.999,120.478 427.447,119.584 428.951,118.63 430.51,117.62 432.121,116.557 433.779,115.444 435.481,114.283 437.225,113.079 439.007,111.835 440.822,110.554 \n",
       "  442.668,109.242 444.54,107.9 446.434,106.535 448.348,105.149 450.276,103.748 452.214,102.335 454.16,100.915 456.108,99.4924 458.054,98.0711 459.995,96.6559 \n",
       "  461.926,95.2509 463.844,93.8607 465.744,92.4893 467.623,91.1411 469.476,89.8202 471.3,88.5305 473.091,87.276 474.846,86.0606 476.56,84.8877 478.231,83.7609 \n",
       "  479.854,82.6836 481.428,81.6588 482.948,80.6896 484.411,79.7786 485.815,78.9285 487.158,78.1416 488.436,77.4199 489.647,76.7653 490.789,76.1795 491.86,75.6638 \n",
       "  492.858,75.2194 493.781,74.8471 494.628,74.5476 495.397,74.3211 496.088,74.1677 496.7,74.0874 497.231,74.0796 497.682,74.1436 498.051,74.2785 498.34,74.483 \n",
       "  498.547,74.7558 498.674,75.095 498.72,75.4987 498.687,75.9648 498.575,76.4908 498.385,77.0741 498.119,77.7119 497.779,78.4011 497.365,79.1385 496.88,79.9208 \n",
       "  496.325,80.7444 495.704,81.6056 495.017,82.5006 494.269,83.4254 493.461,84.376 492.596,85.3481 491.678,86.3377 490.708,87.3403 489.692,88.3517 488.632,89.3674 \n",
       "  487.531,90.3832 486.392,91.3945 485.221,92.3971 484.02,93.3865 482.793,94.3586 481.545,95.309 480.278,96.2336 478.997,97.1284 477.706,97.9894 476.41,98.8127 \n",
       "  475.111,99.5948 473.815,100.332 472.525,101.021 471.245,101.658 469.979,102.241 468.732,102.767 467.506,103.233 466.307,103.636 465.138,103.975 464.002,104.247 \n",
       "  462.903,104.451 461.845,104.586 460.832,104.649 459.866,104.641 458.95,104.56 458.089,104.406 457.285,104.179 456.54,103.879 455.858,103.507 455.24,103.062 \n",
       "  454.69,102.546 454.209,101.96 453.8,101.305 453.464,100.582 453.202,99.7951 453.018,98.9446 452.911,98.0333 452.882,97.0637 452.933,96.0386 453.065,94.961 \n",
       "  453.277,93.8339 453.571,92.6608 453.945,91.445 454.401,90.1903 454.937,88.9005 455.554,87.5793 456.25,86.231 457.024,84.8595 457.876,83.4692 458.804,82.0641 \n",
       "  459.807,80.6488 460.882,79.2275 462.028,77.8047 463.244,76.3847 464.526,74.9719 465.872,73.5706 467.28,72.1852 468.747,70.8199 470.27,69.479 471.847,68.1664 \n",
       "  473.474,66.8862 475.147,65.6423 476.864,64.4384 478.621,63.2781 480.414,62.1648 482.24,61.1019 484.095,60.0924 485.975,59.1393 487.877,58.2451 489.795,57.4125 \n",
       "  491.727,56.6436 493.669,55.9406 495.615,55.3051 497.563,54.7389 499.508,54.243 501.446,53.8188 503.373,53.4668 505.286,53.1877 507.179,52.9817 509.05,52.8489 \n",
       "  510.893,52.7889 512.707,52.8013 514.486,52.8853 516.228,53.0398 517.928,53.2636 519.583,53.5552 521.19,53.9127 522.745,54.3341 524.247,54.8172 525.691,55.3596 \n",
       "  527.075,55.9584 528.396,56.6109 529.651,57.314 530.839,58.0644 531.958,58.8586 533.004,59.6931 533.977,60.5642 534.875,61.4679 535.696,62.4003 536.439,63.3574 \n",
       "  537.104,64.3348 537.688,65.3285 538.193,66.334 538.616,67.3471 538.958,68.3632 539.219,69.3782 539.399,70.3875 539.499,71.3868 539.518,72.3718 539.459,73.3383 \n",
       "  539.32,74.2819 539.105,75.1985 538.814,76.0842 538.448,76.9351 538.01,77.7472 537.501,78.517 536.924,79.241 536.281,79.9158 535.573,80.5383 534.804,81.1054 \n",
       "  533.977,81.6145 533.094,82.063 532.158,82.4484 531.172,82.7688 530.141,83.0222 529.066,83.2069 527.952,83.3216 526.803,83.3652 525.621,83.3366 524.411,83.2354 \n",
       "  523.176,83.0612 521.921,82.8137 520.649,82.4933 519.365,82.1002 518.071,81.6353 516.774,81.0994 515.475,80.4938 514.181,79.8198 512.893,79.0793 511.618,78.274 \n",
       "  510.358,77.4062 509.117,76.4783 507.9,75.4929 506.711,74.4528 505.552,73.3609 504.428,72.2206 503.343,71.0352 502.3,69.8081 501.302,68.5432 500.353,67.2442 \n",
       "  499.455,65.915 498.613,64.5598 497.828,63.1826 497.104,61.7877 496.443,60.3793 495.848,58.9619 495.321,57.5397 494.864,56.1173 494.479,54.6988 494.168,53.2888 \n",
       "  493.932,51.8916 493.774,50.5114 493.693,49.1525 493.691,47.8191 493.769,46.5152 493.928,45.2448 494.168,44.0117 494.488,42.8196 494.89,41.6722 495.373,40.5727 \n",
       "  495.936,39.5245 496.58,38.5305 497.302,37.5936 498.103,36.7164 498.98,35.9015 499.934,35.1508 500.961,34.4666 502.06,33.8504 503.23,33.3037 504.468,32.8279 \n",
       "  505.772,32.4238 507.139,32.0922 508.567,31.8336 510.054,31.6481 511.595,31.5357 513.189,31.4961 514.832,31.5286 516.52,31.6325 518.251,31.8066 520.021,32.0496 \n",
       "  521.825,32.3598 523.662,32.7354 525.525,33.1744 527.413,33.6743 529.321,34.2328 531.244,34.847 533.18,35.514 535.124,36.2306 537.071,36.9936 539.018,37.7995 \n",
       "  540.961,38.6447 542.896,39.5253 544.819,40.4375 546.725,41.3772 548.611,42.3404 550.473,43.3228 552.308,44.3202 554.11,45.3283 555.877,46.3427 557.605,47.359 \n",
       "  559.29,48.3728 560.929,49.3797 562.519,50.3755 564.057,51.3557 565.54,52.3161 566.964,53.2526 568.327,54.161 569.626,55.0373 570.859,55.8777 572.024,56.6783 \n",
       "  573.118,57.4356 574.14,58.1462 575.088,58.8066 575.96,59.4139 576.755,59.965 577.472,60.4574 578.11,60.8884 578.667,61.2558 579.145,61.5576 579.541,61.7919 \n",
       "  579.856,61.9572 580.089,62.052 580.242,62.0754 580.315,62.0266 580.308,61.9049 580.221,61.7102 580.057,61.4422 579.816,61.1014 579.5,60.6881 579.11,60.2032 \n",
       "  578.648,59.6477 578.115,59.0227 577.516,58.3299 576.85,57.571 576.122,56.748 575.333,55.8631 574.486,54.9188 573.585,53.9178 572.632,52.8628 571.631,51.7571 \n",
       "  570.585,50.6038 569.496,49.4063 568.37,48.1683 567.209,46.8934 566.018,45.5855 564.799,44.2486 563.557,42.8868 562.296,41.5042 561.019,40.1051 559.731,38.6938 \n",
       "  558.436,37.2746 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 3\n",
    "t = 0:0.01:5*pi;\n",
    "x = t+sin.(5*t);\n",
    "y = t+sin.(6*t);\n",
    "plot(x,y)\n",
    "plot!(title= \"A Non-function Parametric Curve\", xlabel = \"x\", ylabel = \"y\", legend = false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We conclude this section with the cycloid, the favorite child of parametric curves as it actually solves certain, easy-to-state problems (see the brachistochrone problem and the tautochrone problem).\n",
    "The defining function for the cycloid is\n",
    "$$x = r(\\theta-\\sin\\theta)\\quad y=r(1-\\cos\\theta)\\quad\\theta\\in\\mathbb{R}$$\n",
    "For now, we will just look at what the cycloid looks like.\n",
    "The code to plot it is below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
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       "  209.57,301.269 209.775,302.056 209.977,302.84 210.176,303.622 210.372,304.401 210.566,305.177 210.757,305.951 210.945,306.722 211.131,307.49 211.314,308.256 \n",
       "  211.494,309.018 211.671,309.777 211.846,310.533 212.019,311.285 212.188,312.035 212.355,312.78 212.52,313.523 212.682,314.262 212.841,314.997 212.998,315.728 \n",
       "  213.152,316.456 213.303,317.18 213.452,317.9 213.599,318.616 213.743,319.328 213.884,320.035 214.024,320.739 214.16,321.438 214.294,322.133 214.426,322.824 \n",
       "  214.556,323.51 214.683,324.191 214.807,324.868 214.929,325.54 215.049,326.208 215.167,326.87 215.282,327.528 215.395,328.181 215.506,328.828 215.614,329.471 \n",
       "  215.72,330.108 215.824,330.741 215.926,331.368 216.026,331.989 216.123,332.606 216.218,333.216 216.311,333.822 216.402,334.421 216.491,335.015 216.578,335.604 \n",
       "  216.663,336.186 216.746,336.763 216.827,337.334 216.905,337.899 216.982,338.457 217.057,339.01 217.13,339.557 217.201,340.098 217.27,340.632 217.337,341.16 \n",
       "  217.403,341.682 217.467,342.197 217.528,342.706 217.588,343.209 217.647,343.705 217.703,344.194 217.758,344.677 217.812,345.153 217.863,345.623 217.913,346.085 \n",
       "  217.961,346.541 218.008,346.99 218.053,347.432 218.097,347.867 218.139,348.296 218.18,348.717 218.219,349.131 218.257,349.538 218.293,349.937 218.328,350.33 \n",
       "  218.362,350.715 218.394,351.093 218.425,351.464 218.454,351.828 218.483,352.184 218.51,352.532 218.536,352.873 218.561,353.207 218.584,353.533 218.607,353.852 \n",
       "  218.628,354.163 218.649,354.466 218.668,354.762 218.686,355.05 218.703,355.33 218.72,355.603 218.735,355.867 218.75,356.124 218.763,356.374 218.776,356.615 \n",
       "  218.788,356.848 218.799,357.074 218.809,357.292 218.819,357.501 218.827,357.703 218.835,357.897 218.843,358.083 218.85,358.261 218.856,358.431 218.862,358.592 \n",
       "  218.867,358.746 218.871,358.892 218.875,359.029 218.879,359.159 218.882,359.28 218.885,359.393 218.887,359.498 218.889,359.595 218.891,359.684 218.892,359.764 \n",
       "  218.893,359.837 218.894,359.901 218.895,359.957 218.895,360.004 218.896,360.044 218.896,360.075 218.896,360.099 218.896,360.114 218.896,360.12 218.896,360.119 \n",
       "  218.896,360.109 218.896,360.091 218.896,360.065 218.897,360.031 218.897,359.988 218.898,359.937 218.898,359.878 218.899,359.811 218.9,359.736 218.902,359.652 \n",
       "  218.904,359.561 218.906,359.461 218.908,359.353 218.911,359.237 218.915,359.113 218.918,358.98 218.923,358.84 218.927,358.691 218.933,358.535 218.938,358.37 \n",
       "  218.945,358.197 218.952,358.016 218.96,357.828 218.968,357.631 218.977,357.426 218.987,357.214 218.997,356.993 219.009,356.765 219.021,356.528 219.034,356.284 \n",
       "  219.048,356.032 219.063,355.772 219.078,355.504 219.095,355.229 219.112,354.946 219.131,354.655 219.151,354.357 219.171,354.051 219.193,353.737 219.216,353.416 \n",
       "  219.24,353.087 219.265,352.75 219.292,352.407 219.319,352.055 219.348,351.697 219.378,351.33 219.41,350.957 219.443,350.576 219.477,350.188 219.512,349.793 \n",
       "  219.549,349.391 219.587,348.981 219.627,348.565 219.668,348.141 219.711,347.71 219.755,347.273 219.801,346.828 219.848,346.377 219.897,345.918 219.948,345.453 \n",
       "  220,344.981 220.054,344.503 220.109,344.018 220.166,343.526 220.225,343.027 220.286,342.522 220.349,342.011 220.413,341.493 220.479,340.969 220.547,340.439 \n",
       "  220.617,339.902 220.689,339.359 220.762,338.81 220.838,338.255 220.915,337.694 220.995,337.127 221.076,336.554 221.16,335.975 221.245,335.391 221.333,334.8 \n",
       "  221.423,334.204 221.514,333.603 221.608,332.995 221.704,332.383 221.802,331.764 221.903,331.141 222.005,330.512 222.11,329.878 222.217,329.238 222.326,328.594 \n",
       "  222.438,327.944 222.552,327.29 222.668,326.63 222.786,325.966 222.907,325.297 223.03,324.623 223.155,323.944 223.283,323.261 223.414,322.574 223.546,321.882 \n",
       "  223.681,321.185 223.819,320.484 223.959,319.779 224.101,319.07 224.246,318.356 224.394,317.639 224.544,316.918 224.696,316.192 224.851,315.463 225.009,314.73 \n",
       "  225.169,313.994 225.332,313.254 225.497,312.51 225.665,311.763 225.836,311.012 226.009,310.259 226.185,309.502 226.363,308.741 226.545,307.978 226.728,307.212 \n",
       "  226.915,306.443 227.104,305.671 227.296,304.896 227.491,304.118 227.688,303.338 227.889,302.556 228.091,301.771 228.297,300.983 228.505,300.193 228.717,299.401 \n",
       "  228.931,298.607 229.147,297.811 229.367,297.013 229.589,296.212 229.814,295.411 230.042,294.607 230.273,293.802 230.507,292.995 230.743,292.186 230.982,291.376 \n",
       "  231.224,290.565 231.469,289.753 231.717,288.939 231.968,288.124 232.221,287.309 232.477,286.492 232.737,285.674 232.999,284.856 233.264,284.037 233.531,283.217 \n",
       "  233.802,282.397 234.076,281.577 234.352,280.756 234.631,279.935 234.913,279.113 235.198,278.292 235.486,277.47 235.777,276.649 236.071,275.827 236.367,275.006 \n",
       "  236.667,274.185 236.969,273.365 237.274,272.545 237.582,271.725 237.893,270.906 238.207,270.088 238.523,269.271 238.843,268.454 239.165,267.639 239.49,266.824 \n",
       "  239.818,266.011 240.149,265.199 240.483,264.388 240.819,263.578 241.159,262.77 241.501,261.963 241.846,261.158 242.194,260.355 242.545,259.553 242.898,258.754 \n",
       "  243.254,257.956 243.613,257.16 243.975,256.366 244.34,255.575 244.707,254.785 245.077,253.998 245.45,253.214 245.826,252.432 246.204,251.652 246.585,250.875 \n",
       "  246.969,250.101 247.356,249.329 247.745,248.561 248.137,247.795 248.531,247.032 248.928,246.273 249.328,245.517 249.731,244.763 250.136,244.014 250.544,243.267 \n",
       "  250.954,242.524 251.367,241.785 251.783,241.049 252.201,240.317 252.621,239.589 253.045,238.864 253.47,238.144 253.899,237.427 254.329,236.714 254.763,236.006 \n",
       "  255.198,235.302 255.637,234.602 256.077,233.906 256.52,233.215 256.966,232.528 257.414,231.846 257.864,231.168 258.316,230.496 258.771,229.827 259.229,229.164 \n",
       "  259.688,228.506 260.15,227.852 260.614,227.203 261.081,226.56 261.55,225.922 262.02,225.289 262.494,224.661 262.969,224.038 263.446,223.421 263.926,222.81 \n",
       "  264.408,222.203 264.892,221.603 265.378,221.008 265.866,220.419 266.356,219.835 266.848,219.258 267.342,218.686 267.838,218.12 268.336,217.56 268.836,217.006 \n",
       "  269.338,216.459 269.842,215.917 270.348,215.382 270.855,214.852 271.365,214.33 271.876,213.813 272.389,213.303 272.904,212.8 273.421,212.303 273.939,211.812 \n",
       "  274.459,211.328 274.981,210.851 275.504,210.381 276.029,209.917 276.556,209.46 277.084,209.01 277.614,208.567 278.145,208.13 278.678,207.701 279.212,207.279 \n",
       "  279.748,206.864 280.285,206.456 280.824,206.055 281.364,205.661 281.905,205.275 282.448,204.895 282.992,204.523 283.537,204.159 284.084,203.802 284.632,203.452 \n",
       "  285.181,203.11 285.731,202.775 286.283,202.447 286.835,202.128 287.389,201.815 287.944,201.511 288.499,201.214 289.056,200.925 289.614,200.643 290.173,200.369 \n",
       "  290.732,200.103 291.293,199.845 291.855,199.595 292.417,199.352 292.98,199.117 293.544,198.89 294.109,198.671 294.675,198.46 295.241,198.257 295.808,198.062 \n",
       "  296.376,197.875 296.944,197.696 297.513,197.525 298.082,197.362 298.652,197.207 299.223,197.06 299.794,196.921 300.365,196.79 300.937,196.668 301.51,196.553 \n",
       "  302.083,196.447 302.656,196.349 303.229,196.259 303.803,196.177 304.377,196.103 304.951,196.038 305.526,195.981 306.101,195.932 306.676,195.891 307.251,195.858 \n",
       "  307.826,195.834 308.401,195.817 308.976,195.809 309.551,195.809 310.126,195.818 310.702,195.834 311.277,195.859 311.852,195.892 312.426,195.934 313.001,195.983 \n",
       "  313.576,196.041 314.15,196.107 314.724,196.181 315.298,196.263 315.871,196.353 316.444,196.452 317.017,196.558 317.59,196.673 318.162,196.796 318.733,196.927 \n",
       "  319.304,197.066 319.875,197.214 320.445,197.369 321.014,197.532 321.583,197.704 322.151,197.883 322.719,198.071 323.286,198.266 323.852,198.47 324.418,198.681 \n",
       "  324.982,198.9 325.546,199.128 326.11,199.363 326.672,199.606 327.233,199.856 327.794,200.115 328.354,200.381 328.912,200.656 329.47,200.937 330.027,201.227 \n",
       "  330.583,201.524 331.137,201.829 331.691,202.142 332.243,202.462 332.795,202.789 333.345,203.125 333.894,203.467 334.442,203.817 334.988,204.175 335.534,204.54 \n",
       "  336.078,204.912 336.62,205.292 337.162,205.678 337.702,206.072 338.24,206.474 338.777,206.882 339.313,207.297 339.847,207.72 340.38,208.15 340.911,208.586 \n",
       "  341.441,209.03 341.969,209.48 342.496,209.937 343.021,210.401 343.544,210.872 344.066,211.35 344.586,211.834 345.104,212.325 345.62,212.822 346.135,213.326 \n",
       "  346.648,213.836 347.159,214.353 347.669,214.876 348.176,215.405 348.682,215.941 349.186,216.483 349.688,217.031 350.188,217.585 350.686,218.145 351.182,218.711 \n",
       "  351.676,219.283 352.168,219.861 352.658,220.445 353.146,221.034 353.632,221.629 354.115,222.23 354.597,222.837 355.077,223.449 355.554,224.066 356.029,224.689 \n",
       "  356.502,225.317 356.973,225.95 357.442,226.589 357.908,227.232 358.372,227.881 358.834,228.535 359.293,229.193 359.75,229.857 360.205,230.525 360.658,231.198 \n",
       "  361.108,231.876 361.556,232.559 362.001,233.246 362.444,233.937 362.885,234.633 363.323,235.333 363.758,236.037 364.191,236.746 364.622,237.459 365.05,238.175 \n",
       "  365.476,238.896 365.899,239.621 366.319,240.349 366.737,241.082 367.153,241.818 367.566,242.557 367.976,243.3 368.384,244.047 368.789,244.797 369.191,245.55 \n",
       "  369.591,246.307 369.988,247.066 370.382,247.829 370.774,248.595 371.163,249.364 371.55,250.135 371.933,250.91 372.314,251.687 372.693,252.466 373.068,253.249 \n",
       "  373.441,254.033 373.811,254.82 374.178,255.61 374.543,256.402 374.904,257.195 375.263,257.991 375.619,258.789 375.973,259.589 376.323,260.391 376.671,261.194 \n",
       "  377.016,261.999 377.358,262.806 377.697,263.614 378.034,264.424 378.367,265.235 378.698,266.047 379.026,266.86 379.351,267.675 379.673,268.49 379.992,269.307 \n",
       "  380.309,270.124 380.623,270.943 380.933,271.762 381.241,272.581 381.546,273.401 381.848,274.222 382.148,275.042 382.444,275.864 382.738,276.685 383.028,277.507 \n",
       "  383.316,278.328 383.601,279.15 383.883,279.971 384.162,280.792 384.438,281.613 384.712,282.434 384.982,283.254 385.25,284.073 385.515,284.892 385.777,285.711 \n",
       "  386.036,286.528 386.292,287.345 386.545,288.16 386.796,288.975 387.043,289.789 387.288,290.601 387.53,291.412 387.769,292.222 388.005,293.03 388.239,293.837 \n",
       "  388.47,294.643 388.697,295.446 388.922,296.248 389.145,297.048 389.364,297.846 389.581,298.642 389.794,299.436 390.005,300.228 390.214,301.018 390.419,301.805 \n",
       "  390.622,302.59 390.822,303.373 391.019,304.153 391.214,304.93 391.406,305.705 391.595,306.477 391.781,307.246 391.965,308.012 392.146,308.775 392.325,309.535 \n",
       "  392.5,310.292 392.674,311.046 392.844,311.796 393.012,312.543 393.177,313.287 393.34,314.027 393.5,314.763 393.657,315.496 393.812,316.225 393.965,316.95 \n",
       "  394.115,317.671 394.262,318.388 394.407,319.101 394.549,319.81 394.689,320.515 394.826,321.216 394.961,321.912 395.094,322.604 395.224,323.292 395.352,323.975 \n",
       "  395.477,324.653 395.6,325.327 395.721,325.996 395.839,326.66 395.955,327.319 396.069,327.973 396.18,328.623 396.289,329.267 396.396,329.906 396.501,330.54 \n",
       "  396.603,331.169 396.704,331.792 396.802,332.41 396.898,333.022 396.991,333.629 397.083,334.231 397.173,334.827 397.26,335.417 397.346,336.001 397.429,336.58 \n",
       "  397.51,337.152 397.59,337.719 397.667,338.28 397.743,338.835 397.816,339.384 397.888,339.926 397.958,340.463 398.026,340.993 398.092,341.516 398.156,342.034 \n",
       "  398.218,342.545 398.279,343.05 398.338,343.548 398.395,344.039 398.45,344.524 398.504,345.002 398.556,345.474 398.607,345.939 398.656,346.397 398.703,346.848 \n",
       "  398.749,347.292 398.793,347.73 398.835,348.16 398.876,348.583 398.916,349 398.954,349.409 398.991,349.811 399.026,350.206 399.061,350.593 399.093,350.974 \n",
       "  399.125,351.347 399.155,351.713 399.183,352.071 399.211,352.422 399.237,352.765 399.263,353.101 399.286,353.43 399.309,353.751 399.331,354.064 399.352,354.37 \n",
       "  399.371,354.668 399.39,354.959 399.408,355.241 399.424,355.517 399.44,355.784 399.454,356.043 399.468,356.295 399.481,356.539 399.493,356.775 399.505,357.003 \n",
       "  399.515,357.223 399.525,357.436 399.534,357.64 399.542,357.836 399.55,358.025 399.557,358.205 399.563,358.377 399.569,358.542 399.575,358.698 399.579,358.846 \n",
       "  399.583,358.986 399.587,359.118 399.591,359.242 399.593,359.358 399.596,359.466 399.598,359.565 399.6,359.656 399.601,359.739 399.602,359.814 399.603,359.881 \n",
       "  399.604,359.94 399.605,359.99 399.605,360.032 399.605,360.066 399.605,360.092 399.606,360.11 399.606,360.119 399.606,360.12 399.606,360.113 399.606,360.098 \n",
       "  399.606,360.074 399.606,360.042 399.606,360.003 399.607,359.954 399.607,359.898 399.608,359.834 399.609,359.761 399.611,359.68 399.613,359.591 399.615,359.494 \n",
       "  399.617,359.388 399.62,359.275 399.623,359.153 399.626,359.023 399.631,358.885 399.635,358.739 399.64,358.585 399.646,358.423 399.652,358.253 399.659,358.075 \n",
       "  399.666,357.889 399.675,357.694 399.683,357.492 399.693,357.282 399.703,357.064 399.714,356.838 399.726,356.604 399.739,356.363 399.753,356.113 399.767,355.856 \n",
       "  399.783,355.591 399.799,355.318 399.816,355.037 399.834,354.749 399.854,354.453 399.874,354.149 399.896,353.838 399.918,353.519 399.942,353.192 399.967,352.858 \n",
       "  399.993,352.517 400.02,352.168 400.048,351.812 400.078,351.448 400.109,351.077 400.141,350.698 400.175,350.313 400.21,349.92 400.247,349.52 400.284,349.113 \n",
       "  400.324,348.698 400.364,348.277 400.407,347.848 400.45,347.413 400.496,346.97 400.542,346.521 400.591,346.065 400.641,345.602 400.692,345.132 400.746,344.656 \n",
       "  400.801,344.173 400.857,343.683 400.916,343.187 400.976,342.684 401.038,342.175 401.102,341.659 401.167,341.137 401.235,340.608 401.304,340.074 401.375,339.533 \n",
       "  401.448,338.986 401.523,338.433 401.6,337.874 401.679,337.308 401.76,336.737 401.842,336.16 401.927,335.578 402.014,334.989 402.103,334.395 402.194,333.795 \n",
       "  402.288,333.189 402.383,332.578 402.48,331.962 402.58,331.34 402.682,330.713 402.786,330.08 402.892,329.443 403.001,328.8 403.112,328.152 403.225,327.499 \n",
       "  403.34,326.841 403.458,326.178 403.578,325.51 403.7,324.838 403.825,324.161 403.952,323.479 404.081,322.793 404.213,322.102 404.347,321.407 404.484,320.708 \n",
       "  404.623,320.004 404.765,319.296 404.909,318.584 405.056,317.868 405.205,317.148 405.357,316.424 405.511,315.696 405.668,314.964 405.827,314.229 405.989,313.49 \n",
       "  406.154,312.747 406.321,312.001 406.491,311.252 406.663,310.499 406.838,309.743 407.016,308.984 407.196,308.222 407.379,307.456 407.565,306.688 407.753,305.917 \n",
       "  407.944,305.143 408.138,304.366 408.335,303.587 408.534,302.805 408.736,302.021 408.941,301.234 409.148,300.445 409.359,299.654 409.572,298.86 409.787,298.065 \n",
       "  410.006,297.267 410.227,296.468 410.452,295.666 410.679,294.863 410.909,294.058 411.141,293.252 411.377,292.444 411.615,291.634 411.856,290.824 412.1,290.012 \n",
       "  412.347,289.198 412.597,288.384 412.849,287.568 413.105,286.752 413.363,285.935 413.624,285.117 413.888,284.298 414.155,283.479 414.425,282.659 414.698,281.838 \n",
       "  414.973,281.017 415.251,280.196 415.533,279.375 415.817,278.553 416.104,277.732 416.394,276.91 416.686,276.089 416.982,275.268 417.28,274.447 417.582,273.626 \n",
       "  417.886,272.806 418.193,271.986 418.503,271.167 418.816,270.349 419.132,269.531 419.45,268.714 419.772,267.898 420.096,267.084 420.423,266.27 420.753,265.457 \n",
       "  421.086,264.646 421.421,263.836 421.76,263.027 422.101,262.22 422.445,261.415 422.792,260.611 423.142,259.809 423.495,259.008 423.85,258.21 424.208,257.413 \n",
       "  424.569,256.619 424.933,255.827 425.299,255.037 425.669,254.249 426.041,253.463 426.415,252.681 426.793,251.9 427.173,251.122 427.556,250.347 427.942,249.575 \n",
       "  428.33,248.805 428.721,248.039 429.115,247.275 429.511,246.515 429.91,245.757 430.312,245.003 430.716,244.252 431.123,243.505 431.533,242.761 431.945,242.02 \n",
       "  432.359,241.283 432.777,240.55 433.197,239.82 433.619,239.094 434.044,238.373 434.471,237.655 434.901,236.941 435.334,236.231 435.769,235.526 436.206,234.824 \n",
       "  436.646,234.127 437.088,233.435 437.533,232.746 437.98,232.063 438.43,231.384 438.881,230.709 439.336,230.04 439.792,229.375 440.251,228.715 440.712,228.06 \n",
       "  441.176,227.41 441.641,226.764 442.109,226.125 442.58,225.49 443.052,224.86 443.527,224.236 444.003,223.617 444.482,223.004 444.963,222.396 445.447,221.794 \n",
       "  445.932,221.197 446.419,220.606 446.909,220.021 447.4,219.441 447.894,218.867 448.389,218.3 448.887,217.738 449.386,217.182 449.887,216.632 450.391,216.089 \n",
       "  450.896,215.551 451.403,215.02 451.912,214.496 452.422,213.977 452.935,213.465 453.449,212.959 453.965,212.46 454.483,211.968 455.003,211.482 455.524,211.002 \n",
       "  456.047,210.53 456.571,210.064 457.097,209.605 457.625,209.152 458.154,208.707 458.685,208.269 459.217,207.837 459.751,207.413 460.287,206.995 460.823,206.585 \n",
       "  461.362,206.182 461.901,205.786 462.442,205.397 462.984,205.015 463.528,204.641 464.073,204.274 464.619,203.915 465.167,203.562 465.715,203.218 466.265,202.881 \n",
       "  466.816,202.551 467.368,202.229 467.922,201.914 468.476,201.607 469.032,201.308 469.588,201.016 470.146,200.732 470.704,200.456 471.263,200.187 471.824,199.927 \n",
       "  472.385,199.674 472.947,199.428 473.51,199.191 474.074,198.962 474.638,198.74 475.204,198.527 475.77,198.321 476.337,198.124 476.904,197.934 477.472,197.752 \n",
       "  478.041,197.579 478.61,197.413 479.18,197.255 479.75,197.106 480.321,196.965 480.893,196.831 481.465,196.706 482.037,196.589 482.61,196.48 483.183,196.379 \n",
       "  483.756,196.287 484.33,196.202 484.904,196.126 485.478,196.058 486.052,195.998 486.627,195.946 487.202,195.903 487.777,195.868 488.352,195.84 488.927,195.822 \n",
       "  489.502,195.811 490.077,195.809 490.653,195.814 491.228,195.828 491.803,195.851 492.378,195.881 492.953,195.92 493.528,195.966 494.102,196.021 494.676,196.085 \n",
       "  495.251,196.156 495.824,196.236 496.398,196.323 496.971,196.419 497.544,196.523 498.117,196.636 498.689,196.756 499.26,196.885 499.832,197.021 500.402,197.166 \n",
       "  500.973,197.319 501.542,197.479 502.111,197.648 502.68,197.825 503.248,198.01 503.815,198.203 504.381,198.404 504.947,198.613 505.512,198.83 506.076,199.054 \n",
       "  506.64,199.287 507.202,199.527 507.764,199.776 508.325,200.032 508.885,200.296 509.444,200.567 510.002,200.847 510.559,201.134 511.115,201.429 511.67,201.731 \n",
       "  512.224,202.041 512.777,202.359 513.329,202.684 513.879,203.017 514.429,203.357 514.977,203.705 515.524,204.06 516.07,204.423 516.614,204.793 517.157,205.17 \n",
       "  517.699,205.554 518.239,205.946 518.778,206.345 519.316,206.751 519.852,207.164 520.387,207.585 520.92,208.012 521.452,208.446 521.982,208.888 522.511,209.336 \n",
       "  523.038,209.791 523.563,210.253 524.087,210.721 524.609,211.197 525.13,211.679 525.649,212.167 526.166,212.663 526.681,213.165 527.194,213.673 527.706,214.187 \n",
       "  528.216,214.709 528.724,215.236 529.231,215.77 529.735,216.31 530.238,216.856 530.738,217.408 531.237,217.966 531.733,218.53 532.228,219.1 532.721,219.676 \n",
       "  533.211,220.258 533.7,220.846 534.186,221.439 534.671,222.038 535.153,222.643 535.633,223.253 536.112,223.869 536.587,224.49 537.061,225.116 537.533,225.748 \n",
       "  538.002,226.385 538.469,227.027 538.934,227.674 539.396,228.326 539.857,228.983 540.314,229.645 540.77,230.312 541.223,230.983 541.674,231.66 542.123,232.341 \n",
       "  542.569,233.026 543.013,233.716 543.454,234.411 543.893,235.109 544.329,235.813 544.763,236.52 545.195,237.231 545.623,237.947 546.05,238.666 546.474,239.39 \n",
       "  546.895,240.117 547.314,240.848 547.73,241.583 548.144,242.321 548.555,243.063 548.964,243.809 549.369,244.558 549.773,245.31 550.173,246.065 550.571,246.824 \n",
       "  550.967,247.586 551.359,248.351 551.749,249.118 552.136,249.889 552.521,250.663 552.903,251.439 553.282,252.218 553.658,252.999 554.032,253.783 554.403,254.569 \n",
       "  554.771,255.358 555.136,256.149 555.499,256.942 555.859,257.738 556.216,258.535 556.57,259.334 556.921,260.135 557.27,260.938 557.616,261.742 557.959,262.549 \n",
       "  558.299,263.356 558.636,264.166 558.971,264.976 559.302,265.788 559.631,266.601 559.957,267.415 560.28,268.231 560.6,269.047 560.918,269.864 561.232,270.682 \n",
       "  561.544,271.501 561.853,272.32 562.159,273.14 562.462,273.96 562.762,274.781 563.059,275.602 563.354,276.423 563.645,277.245 563.934,278.066 564.22,278.888 \n",
       "  564.503,279.709 564.783,280.531 565.06,281.352 565.334,282.172 565.606,282.993 565.874,283.812 566.14,284.632 566.403,285.45 566.663,286.268 566.92,287.085 \n",
       "  567.174,287.901 567.426,288.716 567.674,289.53 567.92,290.343 568.163,291.154 568.403,291.964 568.64,292.773 568.874,293.58 569.106,294.386 569.335,295.19 \n",
       "  569.56,295.993 569.784,296.793 570.004,297.592 570.221,298.389 570.436,299.184 570.648,299.976 570.857,300.767 571.064,301.555 571.267,302.341 571.468,303.124 \n",
       "  571.666,303.905 571.862,304.683 572.054,305.459 572.244,306.231 572.432,307.001 572.616,307.768 572.798,308.533 572.978,309.294 573.154,310.051 573.328,310.806 \n",
       "  573.499,311.558 573.668,312.306 573.834,313.05 573.998,313.791 574.159,314.529 574.317,315.263 574.473,315.993 574.626,316.719 574.777,317.442 574.925,318.16 \n",
       "  575.07,318.875 575.213,319.585 575.354,320.291 575.492,320.993 575.628,321.691 575.761,322.384 575.892,323.073 576.021,323.758 576.147,324.437 576.271,325.113 \n",
       "  576.392,325.783 576.511,326.449 576.628,327.109 576.742,327.765 576.854,328.416 576.964,329.062 577.072,329.703 577.177,330.339 577.28,330.969 577.381,331.594 \n",
       "  577.48,332.214 577.577,332.828 577.671,333.437 577.764,334.04 577.854,334.637 577.942,335.229 578.028,335.816 578.112,336.396 578.194,336.971 578.274,337.539 \n",
       "  578.352,338.102 578.428,338.659 578.503,339.209 578.575,339.754 578.645,340.292 578.714,340.824 578.78,341.35 578.845,341.87 578.908,342.383 578.969,342.89 \n",
       "  579.029,343.39 579.086,343.883 579.142,344.37 579.197,344.851 579.249,345.325 579.3,345.791 579.35,346.252 579.397,346.705 579.444,347.152 579.488,347.591 \n",
       "  579.531,348.024 579.573,348.449 579.613,348.868 579.652,349.279 579.689,349.684 579.725,350.081 579.759,350.471 579.792,350.853 579.824,351.229 579.855,351.597 \n",
       "  579.884,351.958 579.912,352.311 579.939,352.657 579.964,352.995 579.988,353.326 580.012,353.65 580.034,353.965 580.055,354.274 580.075,354.574 580.094,354.867 \n",
       "  580.111,355.152 580.128,355.43 580.144,355.7 580.159,355.961 580.173,356.216 580.187,356.462 580.199,356.701 580.211,356.931 580.221,357.154 580.231,357.369 \n",
       "  580.241,357.576 580.249,357.775 580.257,357.965 580.264,358.148 580.271,358.323 580.277,358.49 580.282,358.649 580.287,358.8 580.292,358.943 580.296,359.077 \n",
       "  580.299,359.204 580.302,359.322 580.305,359.432 580.307,359.534 580.309,359.628 580.31,359.714 580.312,359.791 580.313,359.861 580.313,359.922 580.314,359.975 \n",
       "  580.314,360.02 580.315,360.056 580.315,360.085 580.315,360.105 580.315,360.117 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The Cycloid\n",
    "r = 1;\n",
    "n = 3;\n",
    "t = 0:0.01:n*2*pi;\n",
    "x = r*(t-sin.(t));\n",
    "y = r*(1-cos.(t));\n",
    "plot(x,y)\n",
    "plot!(title= \"$n Cycles of the Cycloid\", xlabel = \"x\", ylabel = \"y\", legend = false)\n",
    "plot!(ylim=[0,4*r])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">! Spoiler Text"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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